Introduction to Gauge Theory Andriy Haydys 22nd October 2019 This is lecture notes for a course given at the PCMI Summer School “Quantum Field The- ory and Manifold Invariants” (July 1 – July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg–Witten gauge theory. However, I tried to present the material in a form, which is suitable for other gauge-theoretic invariants too. Contents 1 Introduction2 2 Bundles and connections4 2.1 Vector bundles....................................4 2.1.1 Basic notions.................................4 2.1.2 Operations on vector bundles.........................5 2.1.3 Sections....................................6 2.1.4 Covariant derivatives.............................6 2.1.5 The curvature.................................8 2.1.6 The gauge group............................... 10 2.2 Principal bundles................................... 11 2.2.1 The frame bundle and the structure group.................. 11 2.2.2 The associated vector bundle......................... 14 2.2.3 Connections on principal bundles...................... 16 2.2.4 The curvature of a connection on a principal bundle............. 19 2.2.5 The gauge group............................... 21 2.3 The Levi–Civita connection.............................. 22 2.4 Classification of U(1) and SU(2) bundles...................... 23 2.4.1 Complex line bundles............................. 24 2.4.2 Quaternionic line bundles.......................... 25 3 The Chern–Weil theory 26 3.1 The Chern–Weil theory................................ 26 3.1.1 The Chern classes.............................. 28 3.2 The Chern–Simons functional............................ 30 3.3 The modui space of flat connections......................... 32 3.3.1 Parallel transport and holonomy....................... 32 1 Introduction to Gauge Theory 3.3.2 The monodromy representation of a flat connection............. 33 4 Dirac operators 34 4.1 Spin groups and Clifford algebras.......................... 34 4.2 Dirac operators.................................... 36 4.3 Spin and Spinc structures............................... 37 4.3.1 On the classification of spinc structures................... 41 4.4 The Weitzenböck formula............................... 41 5 Linear elliptic operators 43 5.1 Sobolev spaces.................................... 43 5.2 Elliptic operators................................... 46 5.3 Elliptic complexes................................... 49 5.3.1 A gauge-theoretic interpretation....................... 51 5.3.2 The de Rham complex............................ 51 6 Fredholm maps 52 6.1 The Kuranishi model and the Sard–Smale theorem................. 52 6.2 The Z=2Z degree................................... 53 6.3 The parametric transversality............................. 55 6.4 The determinant line bundle............................. 56 6.5 Orientations and the Z–valued degree........................ 58 6.6 An equivariant setup................................. 59 7 The Seiberg–Witten gauge theory 61 7.1 The Seiberg–Witten equations............................ 61 7.1.1 The gauge group action............................ 62 7.1.2 The deformation complex.......................... 62 7.1.3 Sobolev completions............................. 63 7.1.4 Compactness of the Seiberg–Witten moduli space.............. 64 7.1.5 Slices..................................... 67 7.1.6 A perturbation................................ 68 7.1.7 Reducible solutions.............................. 69 7.1.8 Orientability of the Seiberg–Witten moduli space.............. 70 7.2 The Seiberg–Witten invariant............................. 71 7.2.1 Sample application of the Seiberg–Witten invariant............. 72 1 Introduction Gauge theory by now is a vast subject with many connections in geometry, analysis, and physics. In these notes I focus on gauge theory as it is used in the construction of manifolds invariants, other uses of gauge theory remain beyond the scope of these notes. The basic scheme of construction invariants of manifolds via gauge theory is quite simple. To be more concrete, let me describe some details. Thus, let be a Lie group acting on a manifold . The common convention, which I will follow, is that actsG on the right, although this is clearly C G 2 Introduction to Gauge Theory nonessential. The quotient := = may fail to be a nice space1 in a few ways, for example due B C G to the presence of points with nontrivial stabilizers. Denote by irr the subspace consisting of C ⊂ C all points with the trivial stabilizer. Then acts on irr and the quotient irr := irr= is better G C B C G behaved. Let me assume that irr is in fact a manifold. Pick a –representation V Band a smooth –equivariant map F : V , where the equivariancy means theG following: G C! F (a g) = g−1 F (a): · · Here V is thought of as a left –module. More to the point, assumingG that 0 V is a regular value of F , we obtain a submanifold −1 2 irr := F (0) irr= irr. Furthermore, assume also d := dim irr < . irr will be M \C G ⊂ B M 1 M referred to as the ‘moduli space’, although the terminology may seem odd at this moment. If irr M is compact and oriented, it has the fundamental class [ irr] Hd( irr; Z). In particular, for any d M 2 B cohomology class η H ( irr; Z) we obtain an integer 2 B Z [ irr]; η = η; M Mirr where η is thought of as a closed form of degree d on irr. This is the ‘invariant’ we are interested in. B One way to construct cohomology classes on irr is as follows. Assume there is a nor- B mal subgroup 0 so that G := = 0 is a Lie group. Then the ‘framed moduli space’ ^ −1 G ⊂ G G G ^ irr := F (0) irr= 0 is equipped with an action of = 0 = G such that irr=G = irr. M ^\C G G G M M In other words, irr can be viewed as a principal G bundle, whose characteristic classes yield the cohomology classesM we are after. More details on this is provided in Section3. While this scheme is clearly very general, the details in each particular case may differ to some extend. The aim of these notes is to explain this basic scheme in some details rather than variations enforced by a concrete setup. As an illustration I consider the Seiberg–Witten theory in dimension four, where this scheme works particularly well. Due to the tight timeframe of the lectures this remains essentially the only example of a gauge–theoretic problem considered in these notes. At present a number of monographs in gauge theory is available, for example [Mor96, Moo96, DK90, FU91, Don02, KM07] just to name a few. All of them are however pretty much specialized to a concrete setting or problem while many features are common to virtually any theory, which is based on counting solutions of non-linear elliptic PDEs, even not necesserily of gauge–theoretic origin. While the presentation here does not differ substantially from the treatment of [Mor96, DK90], I tried to keep separate general principles from peculiar features of concrete problems. Also, I hopefully streamlined somewhat some arguments, for example the proof of the compactness for the Seiberg–Witten moduli space. Although I did not cover any other examples of gauge– theoretic invariants except the Seiberg–Witten invariant, the reader will be hopefully well prepared to read advanced texts on other enumerative invariants such as [DK90] on his own. The reader may ask why should we actually care about gauge theories beyond the Seiberg– Witten one, since most of the results obtained by gauge–theoretic means can be reproduced with the help of the Seiberg–Witten gauge theory. One answer is that the Seiberg–Witten theory does not generalize to higher dimensions, whereas for example the anti–self–duality equations do generalize 1Ideally, one wishes the quotient to be a smooth manifold. 3 Introduction to Gauge Theory and are currently a topic of intense research. Even in low dimensions physics suggests that gauge theories based on non-abelian groups such as SL(2; C) or more traditional SU(2) may provide insights that the Seiberg–Witten theory is not capable of. For example, the Kapustin–Witten equa- tions play a central rôle in Witten’s approach to the construction of the Jones polynomial. These notes are organized as follows. In introductory Sections2–4 I describe those properties of the ‘points’ in that are important for the intended applications. Sections5 and6 are devoted to analytic tools usedC in the studies of gauge–theoretic moduli spaces. This can be viewed as studies of properties of F from a (somewhat) abstract point of view. The concepts and theorems considered in this part are not really specific to gauge theory, the same toolbox is used in virtually any enumerative problem based on elliptic PDEs. The last section is devoted to the Seiberg–Witten theory. I tried to keep these notes self-contained where this did not require long and technical detours. I hope this will make the notes available to early graduate students. Part of such course is necessarily certain tools from PDEs. I tried to state clearly the facts that play a rôle in the main part of these notes, however I did not intend to give a detailed description or complete proofs of those. The reader may wish to consult a more specialized literature on this topic, for example relevant chapters of [Eva10, Wel80]. 2 Bundles and connections 2.1 Vector bundles 2.1.1 Basic notions Roughly speaking, a vector bundle is just a family of vector spaces parametrized by points of a manifold (or, more generally, of a topological space). More formally, the notion of a vector bundle is defined as follows. Definition 1. Choose a non-negative integer k. A real smooth vector bundle of rank k is a triple (π; E; M) such that the following holds: (i) E and M are smooth manifolds, π : E M is a smooth submersion (the differential is surjective at each point); ! −1 k (ii) For each m M the fiber Em := π (m) has the structure of a vector space and Em = R ; 2 ∼ (iii) For each m M there is a neighborhood U m and a smooth map U such that the following diagram2 3 1 ψU k π− (U) U R × π pr1 U commutes.